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Percolation and Polymer-based Nanocomposites

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Title: Percolation and Polymer-based Nanocomposites


1
Percolation and Polymer-based Nanocomposites
Avik P. Chatterjee Department of Chemistry State
University of New York College of Environmental
Science Forestry
2
Outline
  • Fiber-based composites percolation the What
    and the Why
  • Elastic moduli in rod-reinforced
    nanocomposites
  • A unified model that integrates percolation
    ideas with effective
  • medium theory
  • Analogy between continuum rod percolation and
    site percolation on
  • a modified Bethe lattice calculation of the
    percolation threshold
  • and percolation and backbone probabilities
  • A simple model for systems with non-random
    spatial
  • distributions for the fibers
  • Future Directions and Acknowledgements

3
Composites Percolation
  • Composites Mixtures of particles/nanoparticles
    (the filler), dispersed
  • (randomly or otherwise) within another phase
    (the matrix)
  • Basic idea to combine integrate desirable
    properties (e.g., low density,
  • high mechanical moduli or conductivity) from
    different classes of material
  • Natural Composites (e.g. bone and wood) often
    exhibit hierarchical
  • structures, varying organizational motifs on
    different scales
  • Wood for the Trees wood remains one of the
    most successful
  • fiber-reinforced composites, and cellulose
    the most widely-occurring
  • biopolymer (annual global production of wood
    1.75 x 109 metric tons)

4
Wood A natural, fiber-reinforced composite
Cell walls layered cellulose microfibrils
(linear chains of glucose residues, degree of
polymerization ? 5000 10000, ? 40-50 w/w of
dry wood depending on species), bound to matrix
of hemicellulose and lignin
R.J. Moon, Nanomaterials in the Forest Products
Industry, McGraw-Hill Yearbook in Science
Technology, p. 226 -229, 2008
5
Cellulose Nanocrystals (I)
  • Cellulose (linear chains of glucose residues),
    bound to matrix of lignin
  • and hemicellulose, comprises ? 40-50 w/w of
    dry wood
  • Individual fibers have major dimensions 1-3
    mm, consisting of spirally
  • wound layers of microfibrils bound to
    lignin-hemicellulose matrix
  • microfibrils contain crystalline domains of
    parallel cellulose chains
  • individual crystalline domains 5-20 nm in
    diameter, 1-2 ?m in length
  • Nanocrystalline domains separable from amorphous
    regions by
  • controlled acid hydrolysis (amorphous regions
    degrade more rapidly)
  • Crystalline domain elastic modulus
    (longitudinal) 150 GPa compare
  • martensitic steel 200 GPa, carbon nanotubes
    103 GPa
  • Suggests possible role for cellulose
    nanocrystals as a renewable, bio-
  • based, low-density, reinforcing filler for
    polymer-based nanocomposites

6
Cellulose Nanocrystals (II)
  • Cellulose microfibrils secreted by certain
    non-photosynthetic bacteria
  • (e.g. Acetobacter xylinum), and form the
    mantle of sea-squirts
  • (tunicates) (e.g. Ciona intestinalis)
  • These highly pure forms are free from
    lignin/hemicelluloses fermentation
  • of glucose a possible microbial route to
    large-scale cellulose production.

Nanocrystalline cellulose whiskers, from acid
hydrolysis of bacterial cellulose. Image courtesy
of Profs. W.T. Winter and M. Roman, Dept. of
Chemistry, SUNY-ESF, and Dept. of Wood Science
and Forest Products at Virginia Tech.
Adult sea-squirts
7
Connectedness Percolation What ?
Percolation The formation of infinite, spanning
clusters of connected (defined by spatial
proximity) particles.
Below percolation threshold
Above percolation threshold
8
Percolation Why? Dramatic Effects on Material
Properties
Impact Toughness
Z. Bartczak, et al., Polymer, 40, 2331, (1999)
Izod impact energy (J/m)
Average ligament thickness (mm)
t Matrix ligament thickness
Toughness of HDPE/Rubber Composites Rubber
particle size range 0.36 0.87 mm
9
Electrical conductivity and Elastic Modulus (I)
  • Polypyrrole-coated cellulose whiskers
    investigated as electrically
  • conducting filler particles (L. Flandin, et
    al., Compos. Sci. Technol., 61, 895,
  • (2001) Mean aspect ratio ? 15)
  • Mechanical reinforcement/modulus enhancement
    (tunicate cellulose
  • whiskers in poly-(S-co-BuA), V. Favier et
    al., Macromolecules, 28, 6365, (1995)
  • Mean aspect ratio ? 70)

10
Electrical conductivity and Elastic Modulus (II)
  • Polystyrene (PS) reinforced with Multi-Walled
    Carbon Nanotubes
  • (MWCNTs) (Diameters 150 200 nm Lengths
    5 10 ?m T 190o C
  • critical volume fraction ? 0.019 A.K. Kota,
    et al., Macromolecules, 40, 7400, (2007))

11
Percolation Thresholds for Rod-like Particles
Percolation threshold for rods, ? ? (D / L),
depends approximately inversely upon aspect
ratio, a result supported by (i) Integral
equation approaches based upon the Connectedness
Ornstein- Zernike equation (X. Wang,
A.P. Chatterjee, J. Chem. Phys., 118, 10787,
(2003) R.H.J. Otten and P. van der
Schoot, Phys. Rev. Lett. 103, 225704, (2009))
(ii) Monte-Carlo simulations of both
finite-diameter as well as
interpenetrable rods (L. Berhan, A.M. Sastry,
Phys. Rev. E, 75, 041120, (2007) M.
Foygel, et al., Phys. Rev. B, 71, 104201,
(2005)) (iii) Excluded volume arguments
(I. Balberg, et al., Phys. Rev. B 30, 3933-3943,
(1984))
12
Elasticity Stress, Strain, Stiffness
  • Stress tensor ? (Energy/Volume)
  • Strain tensor ? (Dimensionless)
  • Stiffness tensor C (Energy/Volume) Fourth
    rank tensor
  • C can have a maximum of 21 independent
    elements/elastic coefficients
  • For an isotropic system, there are only TWO
    independent elastic
  • constants, usually chosen from amongst E,
    G, K, ? (tensile, shear,
  • and bulk modulus, and Poisson ratio,
    respectively)
  • In terms of deformation energy per unit volume,
    U
  • and
    , where

13
A Seismic Interlude
  • Seismic (earthquake-generated) waves
  • Primary/Pressure (P-waves)
  • Longitudinal
  • ? 5-7 km/s in crust, ? 8 km/s in mantle
  • Secondary/Shear (S-waves)
  • Transverse
  • ? 3-4 km/s in crust, ? 5 km/s in mantle
  • Cannot traverse liquid outer core of the
    earth
  • Shadow zones and travel times reveal
    information regarding
  • internal structure of the earth

(Courtesy US Geological Survey)
14
Elastic Moduli of Composite Materials
  • Simple estimates employing only (i) the volume
    fractions ?n, and (ii) the
  • elastic coefficients En, of the individual
    constituents, include the Voigt-
  • Reuss and Hashin-Shtrikman bounds (for
    isotropic materials) (Z. Hashin, S.
  • Shtrikman, J. Mech. Phys. Solids, 11, 127,
    (1963))
  • Arithmetic mean (Hill average) of such bounds
    frequently employed as a
  • semi-empirical tool (R. Hill, Proc. Phys. Soc.
    London A, 65, 349, (1952) S. Ji, et al., J.
  • Struct. Geol., 26, 1377, (2004))

Voigt / Parallel
Reuss / Series
15
Percolation and Elastic Moduli
  • Cellulose nanocrystals modeled as
  • circular cylinders, uniform radius R,
  • variable lengths L
  • Nanocrystals modeled as being
  • transversely isotropic, with five independent
  • elastic constants Eax, Etr, Gtr, ?ax, ?tr
  • Eax 130 GPa, Etr 15 GPa, Gtr 5 GPa, ?ax
    ?tr 0.3
  • (L. Chazeau, et al., J. Appl. Polym. Sci., 71,
    1797, (1999))
  • Actual unit cell symmetry monoclinic, implying
    13 independent elastic
  • constants (K. Tashiro, M. Kobayashi, Polymer,
    32, 1516, (1991))
  • Objective To unify percolation ideas with
    effective medium theory towards
  • an integrated model for
    composite properties

2 R
L
16
Model for Network Contribution to Elasticity
  • For network element of length L and radius R
    elastic deformation energy U is
  • Combines stretching, bending, shearing energies,
    ?ij are
  • strain components in fiber-fixed frame, with
    fiber axis in the
  • Z-direction (F. Pampaloni, et al., Proc. Natl.
    Acad. Sci. US., 103,
  • 10248, (2006))
  • Assume (i) isotropic orientational
    distribution,
  • (ii) random contacts between rods, Poisson
    distribution for lengths of network
  • elements, and (iii) affine deformation
  • Energy of elastic deformation averaged over rod
    and segment lengths and
  • orientations, strain tensor transformed to
    laboratory frame, then differentiated twice
  • with respect to ?ij to obtain estimates
    for the network moduli Enet, Gnet
  • BUT Rods of different lengths will differ in
    likelihood of belonging to network, and
  • for small enough volume fractions, no
    network exists !

17
Percolation Probability and Threshold
  • Percolation probability, denoted P (?, L)
    probability that a randomly
  • selected rod of length L belongs to the
    percolating network/infinite cluster
  • Modeled as
    for , and zero otherwise
  • , L-dependent
    percolation threshold
  • Parameters are treated as
    adjustable control transition width
  • location of threshold
  • These assumptions, together with
    piecewise-linear (unimodal) model for
  • the overall distribution over rod lengths L,
    allows estimation of
  • (i) volume fraction of rods belonging to
    network (?net),
  • (ii) volume fraction and length distribution
    of rods that remain dispersed,
  • (iii) average length of network elements
    (assuming random contacts)
  • There remains the task of combining the network
    moduli with contributions
  • from the matrix, and from the dispersed rods

18
Moduli for the Composite
  • Identify a continuum surrogate for network,
    using Swiss cheese analogy, where
  • (matrix dispersed rods) ? (spherical voids)
  • Use Mori-Tanaka (MT) model to estimate moduli
    for an isotropic system made up of
  • (matrix dispersed rods) (Dispersed rods
    treated as length-polydisperse, within a discrete
  • two-component description) (Y.P. Qiu, G.J.
    Weng, Int. J. Eng. Sci., 28, 1121, (1990))
  • Final step again use Mori-Tanaka (MT) method to
    estimate moduli for the system
  • (continuum network surrogate) (spherical
    voids now filled with isotropic system of
  • matrix dispersed rods, with moduli
    determined in previous step)
  • Recursive use of MT model equivalent in this
    context to Hashin-Shtrikman upper
  • bound

19
Results (I)
  • Matrix Copolymer of ethylene oxide
    epichlorohydrin (EO-EPI)
  • Filler Tunicate cellulose whiskers (solution
    cast system)
  • Mean whisker radius R 13.35 nm
  • Whisker Lengths Ln 2.23 ? m, Lmax 5.37 ? m
    (reproduces first two
  • moments of experimentally measured
    distribution)
  • Solid Line Polydispersity in rod lengths
    included in model
  • Broken Line Model treats rods as monodisperse,

J.R. Capadona, et al., Nat. Nanotechnol., 2, 765,
(2007) D.A. Prokhorova, A.P. Chatterjee,
Biomacromolecules, 10, 3259, (2009)
20
Results (II)
  • Matrix waterborne polyurethane (WPU)
  • Filler Flax cellulose nanocrystals
  • Mean whisker radius R 10.5 nm
  • Whisker Lengths Ln 327 nm, Lmax 500 nm
    (based on experimentally
  • measured distribution of rod lengths)
  • Solid Line Polydispersity in rod lengths
    included in model
  • Broken Line Model treats rods as monodisperse,

X. Cao, et al., Biomacromolecules, 8, 899,
(2007) D.A. Prokhorova, A.P. Chatterjee,
Biomacromolecules, 10, 3259, (2009)
21
Results (III)
  • Matrix Copolymer of styrene butyl acrylate
    (poly-(S-co-BuA))
  • Filler Tunicate cellulose whiskers
  • Mean whisker radius R 7.5 nm
  • Whisker Lengths Ln 1.17 ? m, Lmax 3.0 ? m
    (based on experimentally
  • measured distribution of rod lengths)
  • Solid Line Polydispersity in rod lengths
    included in model
  • Broken Line Model treats rods as monodisperse,

M.A.S.A. Samir, F. Alloin, A. Dufresne,
Biomacromolecules, 6, 612, (2005) D.A.
Prokhorova, A.P. Chatterjee, Biomacromolecules,
10, 3259, (2009)
22
Can Continuum Rod Percolation be related to
Percolation on the Bethe Lattice ?
  • Perfect dendrimer/Cayley tree, with uniform
  • degree of branching z at each vertex
  • No loops/closed paths available
  • Extensively studied as exemplar of
  • mean-field lattice percolation
  • (M.E. Fisher, J.W. Essam, J. Math. Phys., 2,
    609,
  • (1961) R.G. Larson, H.T. Davis, J. Phys. C,
    15, 2327,
  • (1982))
  • If vertices are occupied with probability
  • ?, then site percolation threshold located
  • at ? (1/(z 1))

Portion of a Bethe lattice with z 3
  • Given how thoroughly this problem has been
    examined,question
  • arises whether one can relate continuum rod
    percolation to
  • percolation on the Bethe lattice

23
Continuum Rod Percolation ? Bethe Lattice A
simple-minded mapping
  • Consider a population of rods, with uniform
    radius R, but variable lengths
  • L. We let denote the distribution
    over rod lengths
  • On average a rod of length L experiences ? ?
    (L/R) contacts with other rods
  • in the system
  • For a Bethe lattice with degree z each occupied
    site has (on average) z ?
  • contacts with nearest neighbor occupied
    sites
  • Suggests the following analogy
  • Rod ? Occupied lattice site
  • ? ?
    ?
  • z ? z (L) ? (L/R)

The corresponding Bethe lattice must have a
distribution of vertex degrees
24
Percolation on a modified Bethe Lattice
  • Bethe Lattice Analog Site percolation on a
  • modified Bethe lattice, with site occupation
  • probability ? , and with vertex degree
  • distribution f (z) that can be obtained from
  • the underlying rod length distribution
  • Let Probability that a randomly chosen
  • branch in such a lattice, for which it is
    known
  • that one of the terminal sites is occupied,
  • does not lead to the infinite cluster
  • Given the absence of closed loops, is
  • determined by (M.E.J. Newman, et al., Phys.
    Rev.
  • E, 64, 026118, (2001) M.E.J. Newman, Phys.
    Rev.
  • Lett., 103, 058701, (2009))

Illustration of a portion of modified Bethe
lattice
where the summation runs over all values of z
(L), and depends only upon ? and f (z)
25
Percolation Threshold and Probability
  • Percolation probability for a rod of length L
    probability that an occupied
  • site with vertex degree z (L) belongs to the
    infinite cluster
  • Percolation threshold value of ? at which a
    solution exists for other
  • than the trivial solution 1, P 0
  • for the case that L gtgt R for all rods in the
    system (A.P. Chatterjee, J. Chem.
  • Phys., 132, 224905, (2010) an identical
    result was derived in the field of scale free
  • (power law) networks some years ago, R.
    Albert, A.-L. Barabasi, Rev. Mod. Phys., 74, 47,
  • (2002))
  • Percolation threshold governed by
    weight-averaged aspect ratio
  • consistent with results from a recent integral
    equation-based study (R.H.J.
  • Otten, P. van der Schoot, Phys. Rev. Lett.,
    103, 225704, (2009))

26
Generalization to finite-diameter rods
  • Model rods as hard core soft shell
  • entities hard core radii and lengths
  • denoted R, L, and soft shell radii and
  • lengths given by R l, L 2 l
  • For this problem, an identical approach yields
  • (A.P. Chatterjee, J. Statistical Physics, 146,
    244, (2012))
  • This is in full agreement with recent findings
    based on integral equation
  • approach, for arbitrarily correlated joint
    distributions over rod radii and
  • lengths (R.H.J. Otten, P. van der Schoot,
    Phys. Rev. Lett. 103, 225704, (2009), J.
  • Chem. Phys. 134, 094902, (2011))
  • Result can be generalized to particles with
    arbitrary cross-sectional
  • shapes, not just circular cylinders (A.P.
    Chatterjee, J. Chem. Phys., 137,
  • 134903, (2012))

27
Network and Backbone Volume Fractions
  • A particle is said to belong to the backbone
    of the network if it
  • experiences at least two contacts with the
    infinite cluster
  • Let B (z (L)) Probability that a rod of length
    L belongs to the backbone
  • Then
    (R.G. Larson, H.T. Davis, J.
    Phys. C,
  • 15, 2327, (1982))
  • Volume fractions occupied by the network, and
    network backbone
  • Near Threshold
  • where Lw , Lz are weight and z-averaged rod
    lengths, respectively
  • (A.P. Chatterjee, J. Chem. Phys., 132,
    224905, (2010))

28
Illustrative Results for Percolation Backbone
Probabilities
  • Polydisperse Rods Unimodal Beta distribution,
    with Lmin 8.33 R,
  • Lmax 800 R, Ln 66.67 R, Lw 100 R 1.5
    Ln, Lz 133.34 R 2 Ln
  • ? Black curves
  • Monodisperse Rods (for comparison) Ln Lw Lz
    100 R, same
  • percolation threshold as the polydisperse rod
    population
  • ? Red curves

Upper Black Curves L Lmax Lower Black Curves
L Lmin
29
Network Backbone Volume Fractions
  • Polydisperse Rods Unimodal Beta distribution,
    with Lmin 8.33 R,
  • Lmax 800 R, Ln 66.67 R, Lw 100 R 1.5
    Ln, Lz 133.34 R 2 Ln
  • ? Black curves
  • Monodisperse Rods (for comparison) Ln Lw Lz
    100 R, same
  • percolation threshold as the polydisperse rod
    population
  • ? Red curves

Network, backbone, volume fractions lower for
polydisperse system
30
Comparison with Monte Carlo Simulations
  • Monte Carlo (MC) simulations
  • for various polydisperse
  • rod systems (B. Nigro et al.,
  • Phys. Rev. Lett., 110, 015701, (2013))
  • MC results show reasonable agreement
  • with theory
  • p Number fraction of long rods
  • for bidisperse systems
  • Symbols MC results, various
  • values of p and distributions
  • of aspect ratios
  • Solid Line Theory based upon modified Bethe
    lattice

31
Average number of Contacts at Threshold
  • Average number of contacts
  • per particle at threshold
  • where z is the vertex degree
  • in the modified Bethe lattice
  • For polydisperse systems,
  • Nc can be smaller than unity
  • MC results show qualitative
  • agreement with this result
  • (B. Nigro et al., Phys. Rev. Lett.,
  • 110, 015701, (2013))
  • p Number fraction of long rods,
  • bidisperse systems

A similar result (Nc lt 1) has been reported for
hyperspheres in high-dimensional spaces (N.
Wagner, I. Balberg, and D. Klein, Phys. Rev. E
74, 011127, (2006))
32
Non-random Spatial Distribution of Particles
Heterogeneity effects
  • Foregoing results assume particles are
    distributed in a spatially
  • random, homogeneous fashion
  • But particle distribution may in reality
    display local clustering or
  • aggregation, apparent volume fraction may
    show statistical non-
  • uniformities/fluctuations
  • VERY SIMPLE microstructural descriptors that
    capture such
  • mesoscale heterogeneity
  • (i) Pore radius distribution, and moments
    thereof, e.g. mean pore
  • radius, and
  • (ii) Chord length distribution, and mean
    chord length in the matrix
  • phase (S. Torquato, Random Heterogeneous
    Materials Microstructure and
  • Macroscopic Properties (Springer, New
    York, 2002))

33
Pore Radius Chord Length Distributions (I)
  • Mean values for the pore radius (ltr gt) and chord
    length (lc) provide
  • simple and physically transparent means for
    quantifying spatial
  • non-randomness
  • Goal to construct as simple a description as
    possible in terms of
  • such variables, and to use these as vehicles
    to examine the impact
  • of spatial heterogeneity upon elastic moduli

34
Pore Radius Chord Length Distributions (II)
  • Starting point Random distribution of rods we
    start with a generalization
  • of the traditional Ogston result for the
    distribution of pore sizes in a fiber
  • network, generalized to partially account for
    fiber impenetrability
  • (A.G. Ogston, Trans. Faraday Soc., 54, 1754,
    (1958) J.C. Bosma and J.A. Wesselingh, J.
  • Chromatography B, 743, 169, (2000) A.P.
    Chatterjee, J. Appl. Phys., 108, 063513, (2010))
  • Pore radius distribution in network
  • of fibers of radius R, volume
  • fraction f
  • Mean and mean-squared pore radii

  • where


35
Comparison to Ogston model (D. Rodbard and A.
Chrambach, Proc. Natl. Acad. Sci., 65, 970,
(1970))
  • In the traditional Ogston model
  • where represents the nominal fiber
    volume fraction
  • Relation between models
  • (M.J. Lazzara, D. Blankschtein, W.M. Deen,
  • J. Coll. Interfac. Sci., 226, 112, (2000))
  • For small enough rod volume fractions
  • and our treatment reduces to that of Ogston
    when f ltlt 1

36
Averaging over Fluctuations in f
  • Simplest possible ansatz bimodal distribution
    over mesoscale fibre
  • volume fractions, represent distribution over
    f by

  • where macroscopic

  • fiber volume fraction
  • This distribution function must be
    interpreted in a strictly operational sense !!!
  • Distribution over pore radii, averaged over
    mesoscopic fluctuations in f

  • where is the result for
    a
  • random distribution of fibers
  • Mean and mean-squared pore radii, and mean chord
    lengths, can then
  • be expressed in terms of


37
Simple estimation of moduli as functions of
non-randomness in f
  • Use the foregoing formalism in inverse manner to
    determine f1, f2 for
  • specified values of ltf gt , and for the mean
    pore radius and chord length
  • Values of lt f gt, f1, f2 then used with Hill
    average (arithmetic mean) of
  • the Reuss-Voigt bounds (R. Hill, Proc. Phys.
    Soc. A 65, 349, (1952) J. Dvorkin, et
  • al., Geophysics 72, 1, (2007)) to estimate
    modulus in terms of lt r gt and lt lcgt

Solid Line Modulus for uniform fiber
distribution (supplied as ansatz) Upper and lower
broken lines Hill average of Reuss-Voigt bounds,
obtained from model when lt r gt and lt lcgt each
exceed their values for a random fiber
distribution by factors of 2 and 3,
respectively (A.P. Chatterjee, J. Phys.
Condensed Matter 23, 155104, (2011))
38
Basic Conclusion
  • Mesoscale fluctuations in f lead to
  • (i) increase in the mean pore radius and chord
    length
  • (ii) reduction in the moduli of the material
    (within the empirical Reuss-Voigt-Hill
  • averaging scheme)
  • Alteration in the mean pore radius/pore radius
    distribution may be
  • expected to also affect transport properties,
    e.g., the diffusion coefficient
  • for a spherical tracer
  • Hindered diffusion model Cylindrical fibers
    (radius R) pose steric
  • obstacles to motion of a spherical tracer
    (radius R0), specific interactions
  • neglected

39
Tracer Diffusion in non-random fiber networks (I)
  • Map the pore size distribution for the
    non-random fiber network from our
  • model that for the cylindrical
    cell model (B. Jonsson, et al., Colloid
  • Polym. Sci. 264, 77, (1986) L. Johansson,
    et al., Macromolecules 24, 6024, (1991))
  • Operational volume fraction distribution
    function y (f) maps onto
  • distribution over radii in the cylindrical
    cell model

40
Tracer Diffusion in non-random fiber networks (II)
  • For cylindrical cell model diffusion constant,
    D, for a spherical tracer of
  • radius R0 in system of fibers of radius R
    satisfies (L. Johansson, et al.,
  • Macromolecules 24, 6024, (1991))
  • where h (a) Distribution over outer radii
    in cell model,
  • and D0 unhindered diffusion constant for
    the same tracer when fiber
  • volume fraction vanishes
  • Mapping procedure using bimodal distribution for
    y (f)
  • permits expressing h (a) in terms
    of mean pore radius mean
  • chord length for the fiber
    network as a whole

41
Qualitative findings
  • Results of accounting in this manner for spatial
    fluctuations in f
  • For randomly distributed fibers our approach
    leads to a value for D that is
  • consistently below DOgston , where DOgston
    arises from combining Ogston
  • result for pore radius distribution the
    cylindrical cell model
  • Quite generally Spatial fluctuations in f are
    predicted to increase D, so long
  • as diffusion is controlled primarily by
    steric/excluded-volume effects

Figure shows D / DOgston for different ratios of
the tracer radius (R0) to fiber radius (R) for
randomly distributed fibers
E1 Exponential Integral
42
Tracer Diffusion Some results (I)
  • Diffusion of Bovine Serum Albumin (BSA) (R0
    3.6 nm) in
  • polyacrylamide gel (R 0.65 nm)
  • Triangles Experiment (J. Tong and J.L.
    Anderson, Biophys. J., 70, 1505, (1996))

Legend 1 Fiber distribution random, steric
effects only 2 Fiber distribution random, steric
as well as hydrodynamic effects included (R.J.
Phillips, Biophys. J., 79, 3350, (2000)) 3
Steric and hydrodynamic factors both included,
but fiber distribution non-random Mean pore
radius and chord length are each 1.5 times larger
than for a random fiber network with equal ltf gt
(A.P. Chatterjee, J. Phys. Condensed Matter, 23,
375103, (2011))
43
Tracer Diffusion Some results (II)
  • Diffusion of Bovine Serum Albumin (BSA) (R0
    3.6 nm) in calcium
  • alginate gels (R 0.36 nm)
  • Triangles Experiment (B. Amsden, Polym. Gels
    Networks, 6, 13, (1998))

Legend 1 Fiber distribution random, steric
effects only 2 Fiber distribution random, steric
as well as hydrodynamic effects included (R.J.
Phillips, Biophys. J., 79, 3350, (2000)) 3
Steric and hydrodynamic factors both included,
but fiber distribution non-random Mean pore
radius and chord length are each 1.5 times larger
than for a random fiber network with equal ltf gt
44
  • Future Directions
  • Modeling frequency-dependent moduli, perhaps by
    way of
  • the viscoelastic correspondence principle
  • Modeling anisotropic systems, where fibers have
    some
  • degree of orientational ordering
  • Generalizations to other particle morphologies,
    e.g., disks
  • and persistent/semiflexible linear particles
  • Generalization of the analogy to the Bethe
    lattice to
  • account for local clustering effects (M.E.J.
    Newman, Phys. Rev. Lett.,
  • 103, 058701, (2009))
  • Unification of Bethe-lattice based percolation
    analyses with
  • simple descriptions of non-random
    microstructures

45
Acknowledgements
From SUNY-ESF Dr. Xiaoling Wang Ms. Darya
Prokhorova Dr. DeAnn Barnhart Profs. W.T. Winter
and I. Cabasso From other institutions Prof.
Alain Dufresne Prof. Henri Chanzy Prof. Laurent
Chazeau Prof. Christoph Weder Prof. Jeffrey
Capadona Prof. George Weng Prof. Paul van der
Schoot Prof. Claudio Grimaldi Prof. Isaac
Balberg Dr. Ronald Otten Dr. Biagio Nigro USDA
CSREES National Research Initiative Competitive
Grants Program USDA CSREES McIntire-Stennis
Program National Science Foundation Research
Foundation of the State University of New York
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